Question

The length of the tangent to a circle from a point P, which is 25 cm away from the center is 24 cm. what is the radius of the circle?

Answer 1

Hint: We know that tangent is always perpendicular to the radius made from that point, using this point we’ll use the property of the right-angled triangle so formed. As we get a right-angled triangle, we use the Pythagoras theorem to find the third side, as the hypotenuse and one of the sides is given, the third side will be our radius of the circle.

Complete step by step Answer:

Given data: Length of tangent$ = 24cm$
The distance of the point P from center$ = 25cm$
Let the center of the circle be ‘O’ and, the point tangent touches the circle be ‘R’

We know that tangents are always perpendicular to the radius made from that point.
Therefore, \[OR \bot PR\] or $\angle R = {90^ \circ }$
OP$ = 25cm$
PR$ = 24cm$
Using Pythagoras theorem
i.e., in a right-angled triangle ABC with right angle at A, $B{C^2} = A{B^2} + C{A^2}$
Therefore, Using Pythagoras theorem in triangle OPR where $\angle R = {90^ \circ }$
$ \Rightarrow O{P^2} = P{R^2} + R{O^2}$
Substituting the value of OP and PR, we get,
$ \Rightarrow {25^2} = {24^2} + P{O^2}$
Separating the unknown term, we get,
$ \Rightarrow P{O^2} = {25^2} - {24^2}$
On squaring we get,
$ \Rightarrow P{O^2} = 625 - 576$
On simplification we get,
$ \Rightarrow P{O^2} = 49$
Taking square root on both sides, we get,
$\therefore PO = 7cm$
As, PO is the radius of the circle,
Therefore, the radius of the circle is 7 cm

Note: Here we have given the tangent of the circle, so let us discuss some properties related to the tangent of a circle
1) A tangent of a circle always touches the circle at a single point.
2) Tangent is always perpendicular to the radius made at the point of tangency.
3) The length of two tangents drawn to a single point to a circle is always equal.